#116459

order	512
nilpotency class	3
isoclinism	isoclinic class of #98848
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

(5,6,7,10,8,9,11,12)(13,14,18,21,19,16,17,27)(15,22,25,23,26,28,24,20);(1,2)(3,4)(13,15,17,24,19,26,18,25)(14,20,21,22,16,23,27,28);(2,4)(6,9)(7,11)(13,16)(14,19)(15,23)(17,27)(18,21)(20,26)(22,25)(24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^c=eh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23655);